Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{q^2 - 4q - 45}{q^2 + 10q + 25}$
First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 4q - 45}{q^2 + 10q + 25} = \dfrac{(q - 9)(q + 5)}{(q + 5)(q + 5)} $ Notice that the term $(q + 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 5)$ gives: $r = \dfrac{q - 9}{q + 5}$ Since we divided by $(q + 5)$, $q \neq -5$. $r = \dfrac{q - 9}{q + 5}; \space q \neq -5$